Now, consider \(I_1,I_2\in \mathcal {M}(\mathbb {R})\)  and fix
\(\tilde{I}_1\le I_1\)  and \(\tilde{I}_2\le I_2\) 
realizing \(W^g(I_1,I_2)\) .
Choose \(\tilde{T}\in P(V_\varphi (\tilde{I_1}),V_\varphi (\tilde{I_2}))\)
such that \(\pi _{13}\# \tilde{T}=\min \lbrace G_{\tilde{I}_1}(x), G_{\tilde{I}_2}(y)\rbrace  \in P^{opt}(\tilde{I_1},\tilde{I_2})\) 
(see Theorem 2.18 and Remark 2.19 of [1]}).
Since \(|\tilde{I}_1|=|\tilde{I}_2|\) , by definition of
\(V_\varphi \)  we get:
\(\int |v-w|\ d\tilde{T}(x,v,y,w) =0.\) 
